A formula for the entropy of the convolution of Gibbs probabilities on the circle. (8th June 2018)

• Record Type: Journal Article
• Authors: Lopes, Artur O
• Is Part Of: Nonlinearity. Volume 31:Number 7(2018:Jul.)
• Page Start: 3441
• Page End: 3459
• Publication Date: 2018-06-08
• Subjects: Gibbs probability -- convolution -- expanding map of the circle -- entropy -- Jacobian of a measure; 37A35 -- 37D35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aac5ab
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Consider the transformation, such that (mod 1), and where S 1 is the unitary circle. Suppose is Hölder continuous and positive, and moreover that, for any, we have that We say that ρ is a Gibbs probability for the Hölder continuous potential, if where is the Ruelle operator for . We call J the Jacobian of ρ . Suppose is the convolution of two Gibbs probabilities and associated, respectively, to and . We show that ν is also Gibbs and its Jacobian is given by . In this case, the entropy is given by the expression For a fixed we consider differentiable variations, , of on the Banach manifold of Gibbs probabilities, where, and we estimate the derivative of the entropy at t = 0. We also present an expression for the Jacobian of the convolution of a Gibbs probability ρ with the invariant probability with support on a periodic orbit of period two. This expression is based on the Jacobian of ρ and two Radon–Nidodym derivatives.